Given $ \overrightarrow{PQ}\perp\overrightarrow{PS}$, $ m \angle QPR = 6x - 52$, and $ m \angle RPS = 3x + 34$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
Solution: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since we are given that $\overrightarrow{PQ}\perp\overrightarrow{PS}$ , we know ${m\angle QPS = 90}$ Substitute in the expressions that were given for each measure: $ {6x - 52} + {3x + 34} = {90}$ Combine like terms: $ 9x - 18 = 90$ Add $18$ to both sides: $ 9x = 108$ Divide both sides by $9$ to find $x$ $ x = 12$ Substitute $12$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 6({12}) - 52$ Simplify: $ {m\angle QPR = 72 - 52}$ So ${m\angle QPR = 20}$.